用沃瑟斯坦几何生成分布，让模型学会生成分布本身。
Wasserstein Flow Matching: Generative modeling over families of distributions
- 在分布空间上构建流匹配，利用最优传输理论建模分布间演化。
- 首次实现高维分布的生成，涵盖高斯与点云数据。
- 适用于图形生成与单细胞组学，适合研究复杂分布结构的学者。
生成建模通常将单一源分布通过简单概率流变换为目标分布。但在计算机图形学和单细胞基因组学中，样本本身可视为分布，传统流匹配忽略了其内在几何结构。我们提出沃瑟斯坦流匹配（WFM），将流匹配推广到分布族上，基于沃瑟斯坦几何。值得注意的是，WFM是首个能在高维空间生成分布的算法，无论分布以解析形式（如高斯）还是经验形式（点云）表示。理论分析表明，沃瑟斯坦测地线构成分布空间上的有效条件流，适合作为流匹配目标。算法结合最优传输理论与注意力机制，在不同计算场景下表现出色：对高斯族使用闭式最优传输路径，对一般分布则采用熵正则估计处理点云。实验成功生成2D/3D形状及来自空间转录组数据的高维细胞微环境。代码已公开于 https://github.com/DoronHav/WassersteinFlowMatching。
原文摘要 · Abstract (English)
Generative modeling typically concerns transporting a single source distribution to a target distribution via simple probability flows. However, in fields like computer graphics and single-cell genomics, samples themselves can be viewed as distributions, where standard flow matching ignores their inherent geometry. We propose Wasserstein flow matching (WFM), which lifts flow matching onto families of distributions using the Wasserstein geometry. Notably, WFM is the first algorithm capable of generating distributions in high dimensions, whether represented analytically (as Gaussians) or empirically (as point-clouds). Our theoretical analysis establishes that Wasserstein geodesics constitute proper conditional flows over the space of distributions, making for a valid FM objective. Our algorithm leverages optimal transport theory and the attention mechanism, demonstrating versatility across computational regimes: exploiting closed-form optimal transport paths for Gaussian families, while using entropic estimates on point-clouds for general distributions. WFM successfully generates both 2D & 3D shapes and high-dimensional cellular microenvironments from spatial transcriptomics data. Code is available at https://github.com/DoronHav/WassersteinFlowMatching .
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